Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Consider the matrices 0 7 0 8 0 0 0 0 1 4 13 -1 7 1 0 3 1 2 -1 25 -3 -19 1 -2 -1 A = 3 -6 1 1 1 and R = 1 4 -1 2 -7 1 -5 10 0 0 1 9 1 4 1 17 34 You may assume that R is the reduced echelon form (RREF) of A. Let ā1,. of A, in left-to-right order. ..., ā7 E R° denote the columns (a) Which of the vectors ā3, ā4, and āz belong to the span of ā1 and ā2? No justification necessary. (b) What is the largest number of linearly independent vectors that can be chosen from among ā1,. No justification necessary. ,ā7? (c) Does āz belong to the span of ā1, đ2, and ā4? If not, explain why not; if so, write āz explicitly as a linear combination of ā1, ã2, and đ4. (d) Are ā1, ā2, ā6, and ā7 linearly independent? If so, explain why; if not, provide a nontrivial linear dependence relation among these four vectors (this means an equation of the form ciải + c2ã2 + Cóã6 + czā7 = 0 with at least one of the coefficients nonzero).

Consider the matrices 0 7 0 8 0 0 0 0 1 4 13 -1 7 1 0 3 1 2 -1 25 -3 -19 1 -2 -1 A = 3 -6 1 1 1 and R = 1 4 -1 2 -7 1 -5 10 0 0 1 9 1 4 1 17 34 You may assume that R is the reduced echelon form (RREF) of A. Let ā1,. of A, in left-to-right order. ..., ā7 E R° denote the columns (a) Which of the vectors ā3, ā4, and āz belong to the span of ā1 and ā2? No justification necessary. (b) What is the largest number of linearly independent vectors that can be chosen from among ā1,. No justification necessary. ,ā7? (c) Does āz belong to the span of ā1, đ2, and ā4? If not, explain why not; if so, write āz explicitly as a linear combination of ā1, ã2, and đ4. (d) Are ā1, ā2, ā6, and ā7 linearly independent? If so, explain why; if not, provide a nontrivial linear dependence relation among these four vectors (this means an equation of the form ciải + c2ã2 + Cóã6 + czā7 = 0 with at least one of the coefficients nonzero).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
100%

Please explain part a,c and d using the answer key I have attached below.

Consider the matrices 2 1 4 13 -1 7 |1 0 3 7 0 8 1 -1 25 -3 -19 1 -2 0 -1 0 0 0 0 0 0 A = 3 -6 1 1 1 9. and R = 1 4 -1 -7 1 -5 2 10 1 9. 1 4 1 17 2 34 0 0 ā7 E R° denote the columns You may assume that R is the reduced echelon form (RREF) of A. Let ā1, of A, in left-to-right order. .... (a) Which of the vectors ā3, ā4, and ā, belong to the span of đ1 and ã2? No justification necessary. (b) What is the largest number of linearly independent vectors that can be chosen from among ả1, . No justification necessary. ,ā7? (c) Does āz belong to the span of đ1, ã2, and ā4? If not, explain why not; if so, write āz explicitly as a linear combination of ā1, ã2, and ā4. (d) Are ā1, ā2, ã6, and ā7 linearly independent? If so, explain why; if not, provide a nontrivial linear dependence relation among these four vectors (this means an equation of the form ciãi + c2d2 + coã6 + c7ã7 with at least one of the coefficients nonzero).
Transcribed Image Text:Consider the matrices 2 1 4 13 -1 7 |1 0 3 7 0 8 1 -1 25 -3 -19 1 -2 0 -1 0 0 0 0 0 0 A = 3 -6 1 1 1 9. and R = 1 4 -1 -7 1 -5 2 10 1 9. 1 4 1 17 2 34 0 0 ā7 E R° denote the columns You may assume that R is the reduced echelon form (RREF) of A. Let ā1, of A, in left-to-right order. .... (a) Which of the vectors ā3, ā4, and ā, belong to the span of đ1 and ã2? No justification necessary. (b) What is the largest number of linearly independent vectors that can be chosen from among ả1, . No justification necessary. ,ā7? (c) Does āz belong to the span of đ1, ã2, and ā4? If not, explain why not; if so, write āz explicitly as a linear combination of ā1, ã2, and ā4. (d) Are ā1, ā2, ã6, and ā7 linearly independent? If so, explain why; if not, provide a nontrivial linear dependence relation among these four vectors (this means an equation of the form ciãi + c2d2 + coã6 + c7ã7 with at least one of the coefficients nonzero).
For part (a), only đz belongs to span{ã1,ã2}; đ4 and āz do not. For part (b), the largest such number is 4, since the matrix has four pivot columns. For part (c), the answer is yes, āz does belong to span{ā1, ã2, đ4}: an explicit representation is āz = 7ã1 – ã2 + 4ã4. For part (d), the answer is no, ā1, d2, ā6, and ā7 are not linearly independent. A nontrivial linear dependence relation among them is 8a1 + Oã2 + 9ã6 – ả7 = 0.
Transcribed Image Text:For part (a), only đz belongs to span{ã1,ã2}; đ4 and āz do not. For part (b), the largest such number is 4, since the matrix has four pivot columns. For part (c), the answer is yes, āz does belong to span{ā1, ã2, đ4}: an explicit representation is āz = 7ã1 – ã2 + 4ã4. For part (d), the answer is no, ā1, d2, ā6, and ā7 are not linearly independent. A nontrivial linear dependence relation among them is 8a1 + Oã2 + 9ã6 – ả7 = 0.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS